Báo cáo toán học: "A Traceability Conjecture for Oriented Graphs" docx

A Traceability Conjecture for Oriented Graphs Marietjie Frick ∗ and Susan A van Aardt † Department of Mathematical Sciences, University of South Africa, South Africa {frickm, vaardsa}@unisa.ac.za Jean E Dunbar Converse College, South Carolina, USA jean.dunbar@converse.edu Morten H Nielsen and Ortrud R Oellermann ‡ Department of Mathematics and Statistics, University of Winnipeg, Canada {m.nielsen, o.oellermann}@uwinnipeg.ca Submitted: Jan 26, 2007; Accepted: Nov 20, 2008; Published: Dec 9, 2008 Mathematics Subject Classification: 05C20, 05C38, 05C15 Abstract A (di)graph G of order n is k-traceable (for some k, 1 ≤ k ≤ n) if every induced sub(di)graph of G of order k is traceable. It follows from Dirac’s degree condition for hamiltonicity that for k ≥ 2 every k-traceable graph of order at least 2k − 1 is hamiltonian. The same is true for strong oriented graphs when k = 2, 3, 4, but not when k ≥ 5. However, we conjecture that for k ≥ 2 every k-traceable oriented graph of order at least 2k − 1 is traceable. The truth of this conjecture would imply the truth of an important special case of the Path Partition Conjecture for Oriented Graphs. In this paper we show the conjecture is true for k ≤ 5 and for certain classes of graphs. In addition we show that every strong k-traceable oriented graph of order at least 6k − 20 is traceable. We also characterize those graphs for which all walkable orientations are k-traceable. Keywords: Longest path, oriented graph, k-traceable, Path Partition Conjecture, Traceability Conjecture. ∗ Supported by the National Research Foundation under Grant number 2053752 † Supported by the National Research Foundation under Grant number TTK2004080300021 ‡ Supported by an NSERC grant CANADA. This work was completed in part at a BIRS Workshop, 06rit126 the electronic journal of combinatorics 15 (2008), #R150 1 1 Introduction Let G be a finite, simple graph with vertex set V (G) and edge set E(G). The number of vertices of G is called its order and the number of edges is called its size and are denoted by n (G) and m(G), respectively. Where no confusion arises we will suppress the G. For any nonempty set X ⊆ V (G), X denotes the subgraph of G induced by X. A graph G containing a cycle (path) through every vertex is said to be hamiltonian (traceable). The detour order of G, denoted by λ(G) (as in [3], [20] and [21]), is the order of a longest path in G. The detour deficiency of G is defined as p(G) = n(G) −λ(G). A graph with detour deficiency p is called p-deficient. Thus a graph is traceable if and only if it is 0-deficient. A graph G of order n is k-traceable (for some k, 1 ≤ k ≤ n) if every induced subgraph of G of order k is traceable. Every graph is 1-traceable, but a graph is 2-traceable if and only if it is complete, and a graph of order n is n-traceable if and only if it is traceable. The above concepts are defined analogously for digraphs. Often, a directed path (directed cycle, directed walk) will simply be called a path (cycle, walk). We use A(D) to denote the arc set of a digraph D. If v is a vertex in a digraph D, we denote the sets of out-neighbours and in-neighbours of v by N + (v) and N − (v) and the cardinalities of these sets by d + (v) and d − (v), respectively. The degree of v in D is defined as deg(v) = d + (v) + d − (v) and the minimum degree of D is δ(D) = min v∈V (D) deg(v). If H is a subdigraph of D, then N + (H) =  v∈V (H) N + (v). If S is a subdigraph of D or a set of vertices in D, we denote the out-neighbours of H that lie in S by N + S (H). Similar notation is used with respect to in-neighbours. A digraph is traceable from (to) x ∈ V (D) if D has a hamiltonian path starting (ending) at x. A digraph D is walkable if it contains a walk that visits every vertex. A digraph D is (dis)connected if its underlying graph is (dis)connected and it is called strong (or strongly connected) if every vertex of D is reachable from every other vertex. Thus a nontrivial digraph D is strong if and only if it contains a closed walk that visits every vertex. A maximal strong subdigraph of a digraph D is called a strong component of D. The components of a digraph D are the components of its underlying graph G. The strong components of a digraph have an acyclic ordering, i.e. they may be labelled D 1 , . . . , D t such that if there is an arc from D i to D j , then i ≤ j (cf. [1], p. 17). An oriented graph is a digraph that is obtained from a simple graph by assigning a direction to each edge. In this paper we concentrate on oriented graphs, though some of our results hold for digraphs in general. Section 3 gives a characterization of those graphs for which all walkable orientations are k-traceable. Thomassen [24] showed that for every k ≥ 42 there exists a k-traceable graph of order k + 1 that is nontraceable. (Such graphs are called hypotraceable.) However, for k ≥ 2 all k-traceable graphs of sufficiently large order are hamiltonian, as shown by the following result. Proposition 1.1 Let k ≥ 2 and suppose G is a is k-traceable graph of order at least 2k − 1. Then G is hamiltonian. the electronic journal of combinatorics 15 (2008), #R150 2 Proof. If δ(G) ≥ n 2 , then G is hamiltonian, by Dirac’s degree condition for hamiltonicity (see [8]). Now suppose G has a vertex x such that deg(x) ≤ n−1 2 . Then |V (G) \N (x)| ≥ k. Now let H be an induced subgraph of G such that x ∈ V (H) ⊆ V (G) \N (x) and n (H) = k Then x is an isolated vertex in H, so H is not traceable and hence G is not k-traceable. For digraphs, the situation is very different, even in the case of strong oriented graphs. In Section 4 we show that the analogue of Proposition 1.1 for strong oriented graphs is true when k = 2, 3, 4, but not when k ≥ 5. In fact, we construct, for every n ≥ 5, a nonhamiltonian strong oriented graph of order n that is k-traceable for every k ∈ {5, . . . , n}. For every k ≥ 6 Gr¨otschel et al. [17] constructed a k-traceable oriented graph of order k + 1 that is nontraceable. However, we show that, for k ≥ 2, every strong k-traceable oriented graph of order at least 6k − 20 is traceable. It is therefore natural to ask: for given k ≥ 2, what is the largest value of n such that there exists a k-traceable oriented graph of order n that is nontraceable? We formulate the following conjecture. Conjecture 1 (The Traceability Conjecture (TC)) For every integer k ≥ 2, every k-traceable oriented graph D of order at least 2k − 1 is traceable. This conjecture was motivated by the Path Partition Conjecture for 1-deficient oriented graphs, which is discussed in Section 2. The TC asserts that every nontraceable oriented graph of order n has a nontraceable induced subdigraph of order k for each k ∈ {2, 3, . . . ,  n 2 }. In Section 5 we prove that the Traceabilty Conjecture holds for certain classes of oriented graphs and that it holds in general for k = 2, 3, 4, 5. 2 Background and motivation Our interest in k-traceable graphs and digraphs arose from investigations into the Path Partition Conjecture (PPC) and its directed versions. We briefly sketch the background of these conjectures. Throughout the paper a and b will denote positive integers. A vertex partition (A, B) of a graph G is an (a, b)-partition if λ(A) ≤ a and λ(B) ≤ b. If G has an (a, b)- partition for every pair (a, b) such that a + b = λ(G), then G is λ-partitionable. The PPC can be formulated as follows. Conjecture 2 (Path Partition Conjecture (PPC)) Every graph G is λ-partitionable. The PPC is a well-known, long-standing conjecture. It originated from a discussion between L. Lov´asz and P. Mih´ok in 1981 and was subsequently treated in the theses [18] and [25]. It first appeared in the literature in 1983, in a paper by Laborde, Payan and Xuong [21]. In [4] it is stated in the language of the theory of hereditary properties of graphs. It is also mentioned in [6]. the electronic journal of combinatorics 15 (2008), #R150 3 The analogous conjecture for digraphs is called the DPPC and its restriction to oriented graphs is called the OPPC. In 1995 Bondy [3] stated a seemingly stronger version of the DPPC, requiring λ(A) = a and λ(B) = b. Results on the PPC and its relationship with other conjectures appear in [5], [9], [10], [11], [13] and [15]. Results on the DPPC appear in Laborde et al. [21], Havet [19], Frick et al. [14], and Bang-Jensen et al. [2]. Every graph may be regarded as a symmetric digraph with the same detour order, by replacing every edge with two oppositely directed arcs, and so the truth of the DPPC would imply the truth of the PPC and the OPPC. However, the relationship between the PPC and the OPPC is not clear. The PPC has been proved for all graphs with detour deficiency p ≤ 3. For p ≥ 4 it has been proved for all p-deficient graphs of order at least 10p 2 − 3p (see [5],[15]). We present here an easy proof for the case p = 1, which relies on Proposition 1.1. Proposition 2.1 Every 1-deficient graph is λ-partitionable. Proof. Let G be a 1-deficient graph of order n and consider a pair of positive integers, a, b such that a + b = λ (G) = n − 1. We assume a ≤ b. Then a + 1 = n − b ≤  n 2  . Since G is nonhamiltonian, it follows from Proposition 1.1 that G has a nontraceable induced subgraph H of order a + 1. But then λ (H) ≤ a and |V (G) \ V (H)| = b, so (V (H) , V (G) \ V (H)) is an (a, b)-partition of G. However, the restriction of the OPPC to 1-deficient oriented graphs has not yet been settled and it seems difficult and interesting enough to be formulated as a separate con- jecture. We call it the OPPC1. Conjecture 3 (OPPC1) Every 1-deficient oriented graph is λ-partitionable. The OPPC1 may be formulated in terms of traceability, as follows. Conjecture 4 (Alternative form of OPPC1) If D is a 1-deficient oriented graph of order n = a + b + 1, then D is not (a + 1)-traceable or D is not (b + 1)-traceable. It is clear from the above formulation that the truth of the Traceabilty Conjecture will imply the truth of the OPPC1. 3 Graphs for which all walkable orientations are k- traceable Gr¨otschel and Harary [16] characterized those graphs for which all strong orientations are hamiltonian. Fink and Lesniak-Foster [12] studied the structure of graphs having the property that all walkable orientations are traceable. They showed that if G is obtained from a complete graph of order at least 4 by deleting the edges of a vertex-disjoint union of paths each of length 1 or 2, then every walkable orientation of D is traceable. Graphs for which all strong orientations are eulerian were characterized in [23]. In view of the next result we are mainly interested in walkable orientations of graphs. the electronic journal of combinatorics 15 (2008), #R150 4 Proposition 3.1 Let D be an oriented graph of order n which is k-traceable for some k ∈ {2, . . . , n}. Then D is walkable. Proof. If D is a nonwalkable oriented graph, then D has two vertices x and y such that no path in D contains both x and y. But then every subdigraph of D containing both x and y is nontraceable. We characterize here those graphs of order n for which all walkable orientations are k-traceable for some k ∈ {2, 3, . . . , n}. Proposition 3.2 The complete graph K n is the only graph G of order n having the fol- lowing properties: (1) G has a walkable orientation and (2) every walkable orientation of G is k-traceable for some k ∈ {2, 3, . . . ,  n 2 }. Proof. Every orientation of K n is a tournament, hence k-traceable for all k ∈ {2, 3, . . . , n}, so K n certainly has the claimed properties. Suppose G = K n is a graph of order n satisfying (1) and (2). Note that n ≥ 4 and therefore n − 2 ≥  n 2 . Let x and y be two independent vertices of G. Since G has a k-traceable orientation for some k ∈ {2, 3, . . . ,  n 2 }, G itself is k- traceable, hence hamiltonian by Proposition 1.1. Therefore, G has a hamiltonian cycle C, and it is easy to construct an orientation D of G in which C is a (directed) hamiltonian cycle of D and with d − (x) = d − (y) = 1. Let x − and y − be the in-neighbours of x and y, respectively, in D. Then D − {x − , y − } is not walkable, and hence not k-traceable for any k ∈ {2, 3, . . . , n − 2}. Hence, D is not k-traceable for any k. This contradicts the assumption that G satisfies (2), so such a G cannot exist. 4 Hamiltonicity and traceability of strong, k-traceable oriented graphs It is well-known that every tournament is traceable and every strong local tournament is hamiltonian [1]. (A digraph is a local tournament if, for every vertex v, each of N − (v) and N + (v) is a tournament.) Chen and Manalastas [7] also proved the following result for strong digraphs. Theorem 4.1 (Chen and Manalastas) If D is a strong digraph with α (D) ≤ 2, then D is traceable. Havet [19] strengthened this result to the following. Theorem 4.2 (Havet) If D is a strong digraph with α (D) = 2 then D has two nonadja- cent vertices that are end vertices of hamiltonian paths in D and two nonadjacent vertices that are initial vertices of hamiltonian paths in D. the electronic journal of combinatorics 15 (2008), #R150 5 We shall often use the following result on the minimum degree of k-traceable oriented graphs. Lemma 4.3 Let D be an oriented graph of order n which is k-traceable for some k ≤ n. Then δ (D) ≥ n − k + 1. Proof. Suppose, to the contrary, that D has a vertex x with deg(x) ≤ n − k. Then |V (D) \ N (x)| ≥ k. Let H be an induced subdigraph of D such that V (H) consist of x together with k − 1 other vertices in V (D) \ N(x). Then H has order k and H is nontraceable. Hence D is not k-traceable. We now prove that for strong oriented graphs the analogue of Proposition 1.1 holds for k = 2, 3, 4. Theorem 4.4 For k = 2, 3 or 4, every strong k-traceable oriented graph of order at least k + 1 is hamiltonian. Proof. If k = 2 or 3, then D is a strong local tournament and hence is hamiltonian. Now let D be a strong 4-traceable oriented graph of order n ≥ 5. Since a strong tournament is hamiltonian, we may assume δ(D) ≤ n − 2 and hence, by Lemma 4.3, δ(D) = n − 3 or n − 2. Suppose first that δ (D) = n − 3. Let x be a vertex of degree n − 3 and let {y, z} = V (D)\N [x]. If v ∈ N + (x), then {x, v, y, z} is traceable, so we may assume, without loss of generality, that yz ∈ A(D). Then every vertex in N + (x) is adjacent to y. Furthermore, if v and w are two distinct vertices in N + (x), then {x, v, w, y, } is traceable, so v and w are adjacent. Thus N + (x) is a tournament and hence has a hamiltonian path P . Similarly, every vertex in N − (x) is adjacent from z and N − (x) has a hamiltonian path Q. Thus xP yzQx is a hamiltonian cycle of D. Now suppose δ(D) = n − 2. Let c be the circumference of D and let C = v 1 v 2 . . . v c v 1 be a longest cycle in D. Suppose c ≤ n − 1 and let x ∈ V (G) \C. Suppose that no vertex of C is adjacent to x. Then all vertices of C (except possibly one) are adjacent from x. Since D is strongly connected there is some shortest path P from C to x. Suppose P is a v i − x path. Then x is adjacent to at least one of v i+1 and v i+2 , where the subscripts are modulo c. Let j ∈ {i + 1, i + 2} be such that xv j is an arc of D. Then, since P is a path of order at least 3, P v j v j+1 . . . v i is a cycle longer than C. So at least one vertex of C is adjacent to x and, similarly, at least one vertex of C is adjacent from x. We may also assume that at least one vertex of C is nonadjacent with x; otherwise, there will be a vertex v j on C such that v j ∈ N − (x) and v j+1 ∈ N + (x), producing a cycle longer than C. Hence, since deg(x) ≥ n − 2, exactly one vertex of C, say v 1 , is nonadjacent with x and N + C (x) = {v 2 , . . . , v  } for some index , 2 ≤  < c; otherwise the maximality of the order of C is contradicted. Further, since deg(v 1 ) ≤ n −2, we note that v 1 is a universal vertex of V (C). Next we note that v 1 and x have a common out-neighbour, v 2 , and a common in- neighbour, v c . If v 1 and x have a common out-neighbour, v k = v 2 , then {v 1 , x, v 2 , v k } is nontraceable. Consequently, v 2 is the only common out-neighbour of x and v 1 . Similarly, the electronic journal of combinatorics 15 (2008), #R150 6 v c is the only common in-neighbour of x and v 1 . Thus if 3 ≤ j ≤ c − 1, then v j ∈ N + (x) if and only if v j ∈ N − (v 1 ). Moreover, since {x, v 1 , v 2 , v c } is traceable, v 2 v c ∈ A(D). Now, if 2 <  < c − 1 then, since l is the largest integer such that v  ∈ N + (x), it follows that v  ∈ N − (v 1 ) and v +1 ∈ N + (v 1 ). But then the cycle xv 2 . . . v  v 1 v +1 v +2 . . . v c x is longer than C. If  = 2, then v 3 ∈ N + (v 1 ) and xv 2 v c v 1 v 3 . . . v c−1 x is a cycle longer than C. It is not difficult to show that c > 3 and so, if  = c − 1, then v c−1 ∈ N − (v 1 ) and v 1 v 2 v c xv 3 . . . v c−1 v 1 is a cycle longer than C. A strong 5-traceable oriented graph need not be hamiltonian. Fig. 1 depicts a strong 5-traceable oriented graph of order 6 that is nonhamiltonian. v v x v v v 5 1 2 3 4 Figure 1: A strong 5-traceable oriented graph of order 6 that is nonhamiltonian. v 1 v 2 v 3 v n-2 v v n n -1 Figure 2: A strong k-traceable oriented graph that is nonhamiltonian. Other nonhamiltonian strong 5-traceable oriented graphs are obtained from the graph in Fig. 1 by adding some or all of the arcs v 5 v 2 , v 5 v 3 and v 4 v 2 . However, v 1 and x remain nonadjacent. Nielsen [22] generalized this construction to prove the following. Theorem 4.5 For every n ≥ 5, there exists a strong nonhamiltonian oriented graph of order n that is k-traceable for every k ∈ {5, 6, . . . , n}. Proof. Let T be a transitive tournament of order n ≥ 5 with source vertex s and sink vertex t. Obtain D from T by removing the arc st and by reversing the arcs of the (unique) hamiltonian path of T . This strong oriented graph D is depicted in Fig. 2 with s = v n and t = v 1 . Suppose D has a hamiltonian cycle C. Then, since v 2 is the only out-neighbour of v 1 , the cycle C contains the arc v 1 v 2 , and hence also the arcs v 2 v 3 , v 3 v 4 , . . . , v n−2 v n−1 and v n−1 v n . But v n is not adjacent to v 1 . Thus D is nonhamiltonian. the electronic journal of combinatorics 15 (2008), #R150 7 Now let k ∈ {5, 6, . . . , n} and let H be a subdigraph of D of order k. If H does not contain both v 1 and v n , then H is a tournament and hence is traceable. Now as- sume v 1 , v n ∈ V (H) and let P = u 1 . . . u k−2 be a hamiltonian path of the tournament H −{v 1 , v n }. Since v n has in-degree one in D and v n is a universal vertex in H −v 1 , either v n u 1 . . . u k−2 or u 1 v n u 2 . . . u k−2 is a hamiltonian path of H − v 1 . Since k ≥ 5, every sub- digraph H − v 1 thus contains a hamiltonian path P  = w 1 . . . w k−1 with v n = w k−2 , w k−1 . Similarly, v 1 has out-degree one, so either w 1 . . . w k−1 v 1 or w 1 . . . w k−2 v 1 w k−1 is a hamil- tonian path in H. The graph D 0 constructed by Whitehead in [26], Theorem 2.3, also has the properties required to prove Theorem 4.5. Although that graph as well as the graphs constructed in Theorem 4.5 are nonhamiltonian, they are traceable. We shall prove that all strong k-traceable oriented graphs of sufficiently large order are traceable. The proof relies on the following result. Theorem 4.6 If D is a k-traceable oriented graph of order at least 6k − 20, then D has independence number at most 2. Proof. Suppose S = {x 1 , x 2 , x 3 } is a set of three independent vertices in D and let W = V (D)\S. Let A i = W \N − (x i ) and B i = W \N + (x i ); i = 1, 2, 3. Then A i ∪ B i = W ; i = 1, 2, 3. Now let i, j be any pair of distinct integers in {1, 2, 3}. If |A i ∩ A j | ≥ k − 3, let H be an induced subdigraph of D, whose vertex set consists of x 1 , x 2 , x 3 and k − 3 vertices of A i ∩ A j . Then H has order k and is nontraceable, since both x i and x j have no in-neighbours in H. Hence |A i ∩ A j | ≤ k − 4. Similarly, |B i ∩ B j | ≤ k − 4. Now suppose |A 1 ∩ B 2 | ≥ 2k − 7. Then, since |A 1 ∩ A 3 | ≤ k − 4, at least k − 3 vertices of A 1 ∩ B 2 are in B 3 , but then |B 2 ∩ B 3 | ≥ k − 3. This contradiction proves that |A 1 ∩ B 2 | ≤ 2k − 8. But A 1 = (A 1 ∩ A 2 ) ∪ (A 1 ∩ B 2 ). Hence |A 1 | ≤ (k − 4) + 2k − 8 = 3k − 12. Similarly, |B 1 | ≤ 3k −12. But V (D) = A 1 ∪B 1 ∪{x 1 , x 2 , x 3 }, so n (D) ≤ (3k − 12)+(3k − 12)+3 = 6k − 21. Theorems 4.1 and 4.6 imply the following: Corollary 4.7 If k ≥ 2 and D is a strong k-traceable oriented graph of order at least 6k − 20, then D is traceable. We now show that the case k ≤ 5 of the TC holds for strong oriented graphs. Corollary 4.8 For each k ∈ {2, 3, 4, 5}, every strong k-traceable oriented graph of order at least 2k − 1 is traceable. Proof. Theorem 4.4 proves the cases k = 2, 3, 4. Corollary 4.7 proves that every strong 5-traceable oriented graph of order at least 10 is traceable. Now let D be a strong 5- traceable oriented graph of order 9. Suppose D is not traceable. Then, by Theorem 4.1, D has an independent set of vertices S = {x 1 , x 2 , x 3 } . Now define the sets A i and B i as in Theorem 4.6. Then it follows from the proof of Theorem 4.6 that |A 1 | ≤ 3 and |B 1 | ≤ 3, so |A 1 | = |B 1 | = 3 and A 1 ∩ B 1 = ∅. Hence, since D is 5-traceable, {x 1 , x 2 } ∪ A 1 has a the electronic journal of combinatorics 15 (2008), #R150 8 hamiltonian path starting at x 1 and {x 1 , x 3 } ∪ B 1 has a hamiltonian path ending at x 1 . Thus D is traceable. Corollary 4.8 implies that the case k ≤ 5 of the TC holds for strong oriented graphs. In Section 5 we shall show that the case k ≤ 5 of the TC holds in general. 5 The Traceability Conjecture In this section we deduce some properties of k-traceable oriented graphs and then use these to prove that the TC holds for certain classes of oriented graphs and also that the TC holds for k ≤ 5. From these results we deduce new results concerning the OPPC1. The following useful result follows immediately from the fact that the strong compo- nents of an oriented graph have an acyclic ordering. Lemma 5.1 If P is a path in a digraph D, then the intersection of P with any strong component of D is either empty or a path. In view of Proposition 3.1 we restrict our attention to walkable oriented graphs when investigating the TC. Suppose D is a walkable oriented graph with h strong components. Then the strong components have a unique acyclic ordering D 1 , . . . , D h such that if there is an arc from D i to D j then i ≤ j and there is at least one arc from D i to D i+1 for i = 1, . . . , h − 1. Throughout the paper we shall label the strong components of a walkable oriented graph D in accordance with this unique acyclic ordering and denote the subdigraph of D induced by all the vertices in the strong components D r , D r+1 , . . . , D s by D s r , i.e. D s r =  s  i=r V (D i )  . Our next result gives a lower bound on the order of a strong component that is not a tournament in a k-traceable oriented graph. Lemma 5.2 Let k ≥ 2 and let D be a k-traceable oriented graph of order n ≥ k with strong components D 1 , . . . , D h . If D i is not a tournament for some i ∈ {1, . . . , h}, then n(D i ) ≥ n − k + 3. Proof. Suppose n(D i ) ≤ n−k +2. Then n(D −V (D i )) ≥ n−(n−k +2) = k −2. Now let H be an induced subdigraph of D such that H contains k − 2 vertices of D − V (D i ) and two nonadjacent vertices of D i . Then it follows from Lemma 5.1 that H is nontraceable, contrary to our assumption that D is k-traceable. Next we consider the structure of k-traceable oriented graphs of sufficiently large order. Lemma 5.3 Let k ≥ 2 and let D be a k-traceable oriented graph of order n ≥ 2k − 5 with strong components D 1 , . . . , D h . Then for every positive integer i ≤ h − 1 at least one of the digraphs D i 1 and D h i+1 is a tournament. the electronic journal of combinatorics 15 (2008), #R150 9 Proof. Suppose, to the contrary, that for some i ≤ h − 1 neither D i 1 nor D h i+1 is a tournament. Since n ≥ 2k − 5, one of D i 1 and D h i+1 , say D i 1 , has at least k − 2 vertices. Let H be an induced subdigraph of D such that H contains k − 2 vertices of D i 1 together with any two nonadjacent vertices of D h i+1 . Then it follows from Lemma 5.1 that H is nontraceable, contrary to the hypothesis. For oriented graphs whose nontrivial strong components are all hamiltonian a slightly stronger result than the TC holds. Theorem 5.4 If k ≥ 2 and D is a k-traceable oriented graph of order n ≥ 2k − 3 such that every nontrivial strong component of D is hamiltonian, then D is traceable. Proof. Let the strong components of D be D 1 , . . . D h . Suppose D is nontraceable. Then, since each D i is hamiltonian or a singleton, it is clear that h ≥ 3. If D h−1 1 is a tournament, then every vertex in D h−1 is an end vertex of some hamiltonian path of D h−1 1 . Since D h is hamiltonian or a singleton, this would imply that D is traceable. Thus D h−1 1 is not a tournament. Let i be the smallest positive integer such that D i 1 is not a tournament. Then i ≤ h−1 and it follows from Lemma 5.3 that D h i+1 is a tournament. Hence i > 1 (otherwise D would be traceable) and D i−1 1 is a tournament by the minimality of i. Since D is walkable, there exist vertices y ∈ V (D i ) and w ∈ V (D i+1 ) such that yw ∈ A(D), and since D i is either hamiltonian or a singleton, y is the endvertex of a hamiltonian path P = x . . . y in D i . Suppose x has an in-neighbour v ∈ V (D i−1 ). Since D i−1 1 and D h i+1 are tournaments, v is the endvertex of a hamiltonian path P  in D i−1 1 and w is the initial vertex of a hamiltonian path P  in D h i+1 and so P  P P  is a hamiltonian path of D, a contradiction. Therefore, some vertex of D i has no in-neighbour in D i−1 , i.e. N + D i (D i−1 ) = V (D i ) and D i is not a singleton. If |N + D i (D i−1 )| ≤ n − k then n(D − N + D i (D i−1 )) ≥ k. Let H be an induced subdigraph of D − N + D i (D i−1 ) such that H has k vertices, of which at least one is in D i−1 and at least one in D i . Since D i−1 has no out-neighbours in V (H) ∩ D i , it follows that H is nontraceable. Hence |N + D i (D i−1 )| ≥ n − k + 1 ≥ (2k − 3) − k + 1 = k − 2. Now let C : v 1 v 2 . . . v c v 1 be a hamiltonian cycle of D i . Then, for every v j ∈ N + C (D i−1 ), its predecessor, v j−1 , on C is not in N − C (D i+1 ). Let H be a subdigraph of D induced by a set of k − 2 of these predecessors, together with one vertex from D i−1 and one from D i+1 . Then H has order k but is nontraceable, since D i+1 has no in-neighbours in V (H) ∩ V (C). We are now ready to prove the TC for k ≤ 5. Corollary 5.5 If k ∈ {2, 3, 4, 5} and D is a k-traceable oriented graph of order at least 2k − 1, then D is traceable. Proof. Suppose, to the contrary, that D is nontraceable. By Theorem 5.4, D has a nontrivial strong component X that is nonhamiltonian. By Corollary 4.8 and Lemma 5.2, n−k+3 ≤ n(X) ≤ n−1. Hence k ≥ 4 and n(X) ≥ n−k +3 ≥ (2k−1)−k+3 ≥ k+2 ≥ 6. It therefore follows from Theorem 4.4 that X is not 4-traceable, so k = 5. Now choose the electronic journal of combinatorics 15 (2008), #R150 10 [...]... the TC holds for all oriented graphs with the property that every two cycles are vertex disjoint In particular, it holds for unicyclic oriented graphs Lemma 4.3 implies that the TC holds (vacuously) for every oriented graph D satisfying δ(D) ≤ n(D) Our next result shows that it also holds for oriented graphs with sufficiently 2 large minimum degree Theorem 5.6 If k ≥ 2 and D is a k-traceable oriented graph... theorem The results in this section provide four new results concerning the OPPC1 Corollary 5.7 The OPPC1 holds for a ≤ 4 Corollary 5.8 Let D be a 1-deficient oriented graph If every nontrivial strong component of D is hamiltonian, then D is λ-partitionable Corollary 5.9 Let D be a 1-deficient oriented graph of order n If δ(D) ≤ δ(D) ≥ n − 2, then D is λ-partitionable the electronic journal of combinatorics... and partitions, JCMCC 31 (1999) 137–149 [10] J.E Dunbar and M Frick, The path partition conjecture is true for claw-free graphs, Discr Math 307 (2007) 1285-1290 [11] J.E Dunbar, M Frick, and F Bullock, Path partitions and maximal Pn -free sets, Discr Math 289 (2004) 145–155 [12] J.F Fink and L Lesniak-Foster, Graphs for which every unilateral orientation is traceable, Ars Combinatoria 9 (1980) 113–118... Dlamini, J Dunbar, and O Oellermann, The directed path partition conjecture, Discuss Math Graph Theory 25 (2005) 331–343 [15] M Frick and I Schiermeyer, An asymptotic result on the path partition conjecture, Electronic J Combin 12 (2005) R48 the electronic journal of combinatorics 15 (2008), #R150 12 [16] M Gr¨tschel and F Harary, The graphs for which all strong orientations are hamilo tonian, J Graph... and D is a k-traceable oriented graph of order n ≥ 2k − 3 such that δ(D) ≥ n − 2, then D is traceable Proof Suppose D is nontraceable Then D is not a tournament and therefore we may assume that δ(D) = n − 2 and hence α(D) = 2 Therefore, from Theorem 4.1, it follows that D is not strong Let D1 , , Dh be the strong components of D By Theorem 5.4 there exists a nontrivial strong component Di , 1 ≤... in x h Since δ(D) = n − 2, x is adjacent to every vertex in Di+1 , and hence Di has a hamiltonian path starting at v1 This implies that no vertex in Di−1 is adjacent to v1 Our assumption on δ(D) therefore implies that n(Di−1 ) = 1 and any other hamiltonian path in Di that ends in x also starts at v1 Since Di is strong, v1 has an in-neighbour in Di and since Di is nonhamiltonian, vni is an out-neighbour... (D2 ), then v is an initial vertex of a hamiltonian path of h D2 But then x and y are both nonadjacent with v, contradicting the fact that α(D) = 2 Hence i = 1 Similarly we can show that i = h It therefore follows from Lemma 5.3 that i−1 h both D1 and Di+1 are tournaments Hence every vertex in Di−1 is an end vertex of a i−1 hamiltonian path of D1 and every vertex in Di+1 is an initial vertex of a hamiltonian... [22] M.H Nielsen, Path-cycle sub(di)graphs in well-structured graphs and digraphs PhD thesis, supervised by J Bang-Jensen, University of Southern Denmark, 2006 [23] O.R Oellermann and H.C Swart, Graphs for which all strong orientations are eulerian, Expositiones Mathematicae 2 (1984) 183–184 [24] C Thomassen, Hypohamiltonian and hypotraceable graphs, Discr Math 9 (1974) 91–96 [25] J Vronka, Vertex sets . k-traceable oriented graph of order n that is nontraceable? We formulate the following conjecture. Conjecture 1 (The Traceability Conjecture (TC)) For every integer k ≥ 2, every k-traceable oriented. of this conjecture would imply the truth of an important special case of the Path Partition Conjecture for Oriented Graphs. In this paper we show the conjecture is true for k ≤ 5 and for certain classes. construct, for every n ≥ 5, a nonhamiltonian strong oriented graph of order n that is k-traceable for every k ∈ {5, . . . , n}. For every k ≥ 6 Gr¨otschel et al. [17] constructed a k-traceable oriented